4
cloud physics cloud microphysicscloud dynamics
0◦C warm clouds
cold clouds0◦C
•
•
•
•
•
•
CReSS
2
44 4
4.1
4.1.1
3 1
qv
qc 100 µm
qr
3 kg kg−1 K2.1
∂ρθ
∂t= Adv.θ + Turb.θ − ρw
∂θ
∂z+
ρLv
CpΠ(CNvc − EVcv − EVrv) (4.1)
∂ρqv
∂t= Adv.qv + Turb.qv − ρ (CNvc − EVcv − EVrv) (4.2)
∂ρqc
∂t= Adv.qc + Turb.qc + ρ (CNvc − EVcv − CNcr − CLcr) (4.3)
∂ρqr
∂t= Adv.qr + Turb.qr + ρ (CNcr + CLcr − EVrv) +
∂
∂z(ρUrqr) (4.4)
Adv.φ Turb.φ Lv
J kg−1 Cp J K kg−1 Π (4.4)qr
CNvc condensationEVcv evaporationEVrv evaporationCNcr
autoconversionCLcr collection
1 0.1 0.5mm drizzle
4.1 45
4.1.2
−CNvc + EVcv
Klemp and Wilhelmson (1978) Soong and Ogura (1973)4.2.5
Tetens qvsw
qvsw Tetens
qvsw = ε610.78
pexp
(17.269
Πθ − 273.16Πθ − 35.86
)(4.5)
ε
CNcr, CLcr
CNcr CLcr Kessler (1969)
CNcr = k1 (qc − a) (4.6)
CLcr = k2qcq0.875r (4.7)
k1 = 0.001 s−1 (4.8)
a = 0.001 kg kg−1 (4.9)
k2 = 2.2 s−1 (4.10)
EVrv
Ogura and Takahashi (1971), Klemp and Wilhelmson (1978)
EVrv =1ρ
(1 − qv /qvsw )C (ρqr)0.525
5.4 × 105 + 2.55 × 106 /(pqvsw)(4.11)
C ventilation factor
C = 1.6 + 124.9 (ρqr)0.2046 (4.12)
46 4
Ur
(4.4) Ur Soong and Ogura (1973)
Ur = 36.34 (ρqr)0.1346
(ρ0
ρ
)(4.13)
ρ0 kg m−3 Ur m s−1
z∗ ζ (2.59)
4.2
4.2.1
• 1 2
•
•
•
• → → →
•
•
•
2
•
•
(1999), Murakami et al. (1994), Murakami (1990)
4.2 47
5
θ θ = θ + θ′
qv
qc
qr 100 µm
qi 100 µm
qs 0.1 g cm−3 1 m s−1
qg 0.4 g cm−3 1 4 m s−1
qh 0.9 g cm−3 10 m s−1
CReSS
Ni
Ns
Ng
K kg kg−1 m−3
g kg−1
4.2.2
2.14.1.1
48 4
∂ρθ
∂t= Adv.θ + Turb.θ − ρw
∂θ
∂z+ ρ (Src.θV + Src.θS + Src.θF ) (4.14)
∂ρqv
∂t= Adv.qv + Turb.qv + ρSrc.qv (4.15)
∂ρqc
∂t= Adv.qc + Turb.qc + ρSrc.qc + ρFall.qc (4.16)
∂ρqr
∂t= Adv.qr + Turb.qr + ρSrc.qr + ρFall.qr (4.17)
∂ρqi
∂t= Adv.qi + Turb.qi + ρSrc.qi + ρFall.qi (4.18)
∂ρqs
∂t= Adv.qs + Turb.qs + ρSrc.qs + ρFall.qs (4.19)
∂ρqg
∂t= Adv.qg + Turb.qg + ρSrc.qg + ρFall.qg (4.20)
v, c, r, i, s, g
x y
Adv.φ
Turb.φ
Src.θV
Src.θS
Src.θF
Src.qx
Fall.qx
∂Ni
∂t= Adv.
Ni
ρ+ Turb.
Ni
ρ+ ρSrc.
Ni
ρ+ ρFall.
Ni
ρ(4.21)
∂Ns
∂t= Adv.
Ns
ρ+ Turb.
Ns
ρ+ ρSrc.
Ns
ρ+ ρFall.
Ns
ρ(4.22)
∂Ng
∂t= Adv.
Ng
ρ+ Turb.
Ng
ρ+ ρSrc.
Ng
ρ+ ρFall.
Ng
ρ(4.23)
i, s, g x y
4.2 49
Adv. Nx/ ρ
Turb. Nx/ ρ
Src. Nx/ ρ
Fall. Nx/ ρ
θ 4.14 Src.θV + Src.θS + Src.θF
Src.θV =Lv
CpΠV Dvr (4.24)
Src.θS =Ls
CpΠ(NUAvi + V Dvi + V Dvs + V Dvg) (4.25)
Src.θF =Lf
CpΠ(NUFci + NUCci + NUHci + CLcs + CLcg + CLri + CLrs + CLrg
−MLic − MLsr − MLgr + FRrg − SHsr − SHgr) (4.26)
qv 4.15 Src.qv
Src.qv = −NUAvi − V Dvr − V Dvi − V Dvs − V Dvg (4.27)
qc 4.16 Src.qc
Src.qc = −NUFci − NUCci − NUHci − CLcr − CLcs − CLcg − CNcr + MLic (4.28)
qr 4.17 Src.qr
Src.qr = V Dvr + CLcr − CLri − CLrs − CLrg + CNcr
+MLsr + MLgr − FRrg + SHsr + SHgr (4.29)
qi 4.18 Src.qi
Src.qi = NUAvi + NUFci + NUCci + NUHci
+V Dvi − CLir − CLis − CLig − CNis − MLic + SPsi + SPgi (4.30)
qs 4.19 Src.qs
Src.qs = −SPsi + V Dvs + CLcs + CLrsαrs + CLis − CLsr (1 − αrs) − CLsg
+CNis − CNsg − MLsr − SHsr (4.31)
qg 4.20 Src.qg
Src.qg = −SPgi + V Dvg + PGg + CLri + CLir + (CLrs + CLsr) (1 − αrs)
+CNsg − MLgr + FRrg − SHgr (4.32)
50 4
Ni
ρ4.21 Src.
Ni
ρ
Src.Ni
ρ=
1mi0
NUAvi +Nc
ρqc(NUFci + NUCci + NUHci) + SPN
si + SPNgi
+Ni
ρqi(V Dvi − CLir − CLis − CLig − MLic) − AGN
i − 1ms0
CNis (4.33)
Ns
ρ4.22 Src.
Ns
ρ
Src.Ns
ρ=
Ns
ρqs(V Dvs − MLsr) − CLN
sr (1 − αrs) − CLNsg − AGN
s +1
ms0CNis − CNN
sg (4.34)
Ng
ρ4.23 Src.
Ng
ρ
Src.Ng
ρ=
Ng
ρqg(V Dvg − MLgr) + CLN
ri + CLNrs (1 − αrs) + CNN
sg + FRNrg (4.35)
Lv,Ls,Lf J kg−1 Cp J K kg−1
Π mi0, ms0 kg4.1 4.2.4
NUAvi deposition or sorption nucleationNUFci condensation-freezing nucleationNUCci contact nucleationNUHci homogeneous nucleationSP 2 secondary nucleation of ice crystalsV D vapor deposition, evaporation and sublimationCL collectionPG graupel producitonAG aggregationCN conversionML meltingFR freezingSH shedding of liquid waterSPN 2 secondary nucleation of ice crystalsCLN collectionAGN aggregationCNN conversionFRN freesingαrs 1 − αrs
4.2 51
-VDvr VDvg
NUA vi VD
viVD
vcC
Ncr
CL
cr ML sr,SH sr
CLrs
ML gr,SH gr
CLri,CL Nri,CL rs,CL Nrs,CL rg,FR rg,FR Nrg
NUF ci,NUC ci,NUH ci
ML ic
CLcs
CL is,CN is
SPsi,SP Nsi
CL cg
CL sr,CL sg,CN sg,CN Nsg
SPgi ,SP
Ngi
CL
ir ,CL
ig
AG Ns
AG Ni
Fall. qr
VD
vs
water vapor (qv)
snow (qs,Ns)
cloud water (qc)
rain water (qr)
cloud ice (qi,Ni)
graupel (qg,Ng)
Fall. qg, Fall.( N
g/ )
Fall. qs, Fall. (N
s/ )
4.1.
(4.24),(4.27),(4.28) V Dvc
4.2.5
4.2.3
Marshall and Palmer (1948)λx y nx0
nx (Dx)︸ ︷︷ ︸m−4
= nx0︸︷︷︸m−4
exp( −λx︸︷︷︸m−1
Dx) (4.36)
Marshall-Palmer 2
2 n!(4.36)
Γ (x)
Γ (x) =
∫ ∞
0
exp (−t) tx−1 dt (4.37)
52 4
x 3
fx (Dx)︸ ︷︷ ︸m−1
=1
Γ(νx)
(Dx
Dnx
)νx−1 1Dnx
exp(− Dx
Dnx
)(4.43)
Dx m Γ (νx) 0 ∞ 1νx Dnx
nx (Dx)︸ ︷︷ ︸m−4
= nxt︸︷︷︸m−3
fx (Dx)︸ ︷︷ ︸m−1
(4.44)
nxt x Dx
Dx =∫ ∞
0
Dx fx (Dx) dDx =Γ (νx + 1)
Γ (νx)Dnx = νxDnx (4.45)
(4.38) PP
∫ ∞
0
DPx fx (Dx) dDx =
Γ (νx + P )Γ (νx)
DPnx (4.46)
Γ (x + 1) = xΓ (x) (4.38)
Γ (1) = 1 (4.39)
x n
Γ (n + 1) = n (n − 1) (n − 2) · · · 2 · 1 · Γ (1) = n! (4.40)
Γ
(1
2
)=
√π (4.41)
3∫ ∞
0
D3x exp (−λxDx) dDx =
1
λ4x
Γ (4) =6
λ4x
(4.42)
3 (1999)
4.2 53
(4.36) (4.43) (4.43)
νx = 1 (4.47)
Dnx =1λx
(4.48)
(4.36) (4.46)
∫ ∞
0
DPx λx exp (−λxDx) dDx =
1λP
x
Γ (P + 1) (4.49)
Dx
Dx =1λx
(4.50)
4
Dc =(
6ρqc
πρwNc
) 13
(4.51)
Di =(
6ρqi
πρiNi
) 13
(4.52)
ρw kg m−3 ρi kg m−3
Nc 1 × 108 m−3
nr (Dr) = nr0 exp (−λrDr) (4.53)
ns (Ds) = ns0 exp (−λsDs) (4.54)
ng (Dg) = ng0 exp (−λgDg) (4.55)
nx x y m−4
8 × 106 m−4
4 (1999), Ikawa and Saito (1991), Murakami et al. (1994), Ikawa et al. (1991), Lin et al. (1983)Ferrier (1994)
54 4
mx (Dx) = αuxDβuxx (4.56)
βux = 3
mx = αuxDβuxnx
Γ (νx + βux)Γ(νx)
(4.57)
Ux (Dx) = αuxDβuxx
(ρ0
ρ
)γux
(4.58)
m−4
UxN = αuxDβuxnx
Γ (νx + βux)Γ(νx)
(ρ0
ρ
)γux
(4.59)
Uxq = αuxDβuxnx
Γ (νx + 3 + βux)Γ (νx + 3)
(ρ0
ρ
)γux
(4.60)
(4.47) (4.48) (4.59)(4.60)
UxN = αuxΓ (1 + βux)
λβuxx
(ρ0
ρ
)γux
(4.61)
Uxq = αuxΓ (4 + βux)
6λβuxx
(ρ0
ρ
)γux
(4.62)
ρ0 kg m−3
(4.53) (4.55) 5 x = r, s, g Nx
Nx =∫ ∞
0
nx0 exp (−λxDx) dDx =nx0
λx(4.63)
5Ikawa and Saito (1991)
4.2 55
x
ρqx =∫ ∞
0
π
6ρxD3
xnx0 exp (−λxDx) dDx =πρxnx0
λ4x
(4.64)
2 λx y nx0
λx =(
πρxNx
ρqx
) 13
(4.65)
nx0 = Nx
(πρxNx
ρqx
) 13
(4.66)
x
UxN =1
Nx
∫ ∞
0
Ux (Dx)nx0 exp (−λxDx) dDx
= αuxΓ (1 + βux)
λβuxx
(ρ0
ρ
)γux
(4.67)
(4.61) x
Uxq =1
ρqx
∫ ∞
0
π
6Ux (Dx)D3
xρxnx0 exp (−λxDx) dDx
= αuxΓ (4 + βux)
6λβuxx
(ρ0
ρ
)γux
(4.68)
(4.62) 4.2.6
y m−4 kg m−3
qc ——— αuc = 2.98 × 107, βuc = 2.0, γuc = 1.0 ρw = 1.0 × 103
qr nr0 = 8.0 × 106 αur = 842, βur = 0.8, γur = 0.5 ρw = 1.0 × 103
qi ——— αui = 700, βui = 1.0, γui = 0.33 ρi = 5.0 × 102
qs ns0 = 1.8 × 106 αus = 17, βus = 0.5, γus = 0.5 ρs = 8.4 × 101
qg ng0 = 1.1 × 106 αug = 124, βug = 0.64, γug = 0.5 ρg = 3.0 × 102
56 4
4.2.4
4.2.54.2.7
1 NUAvi, NUFci, NUCci, NUHci
1
→→ NUHci
→ NUAvi
→ NUCci
→ NUFci
NUAvi, NUFci, NUCci, NUHci
1 NUAvi
a 6
Ts w ≤ 0 m s−1
NUAvi =mi0
ρβ2Ni0 exp (β2Ts)
(Si − 1Swi − 1
)B∂Ts
∂zw (4.69)
NUANvi =
NUAvi
mi0(4.70)
(2.59) z∗ ζ
b 7
SSi w ≤ 0 m s−1
NUAvi =mi0
ρ15.25 exp (5.17 + 15.25SSi)
∂SSi
∂zw (4.71)
NUANvi =
NUAvi
mi0(4.72)
(a) z∗ ζ
6Ikawa and Saito (1991), Cotton et al. (1986), Murakami (1990), Ikawa et al. (1991), Murakami et al. (1994), (1999)7Meyers et al. (1992), (1999)
4.2 57
c 8
Ferrier (1994) −5 ◦C Murakami (1990), Cottonet al. (1986) Meyers et al. (1992)
w ≤ 0 m s−1
NUAvi =mi0
ρw
∂Ni
∂z(4.73)
NUANvi =
NUAvi
mi0(4.74)
Ni −5 ◦C
Ni =
⎧⎪⎪⎨⎪⎪⎩
Ni01 exp (β2Ts)(
Si − 1Swi − 1
)B
, T ≥ −5 ◦C
Ni02 exp (a1SSi − b1) , T < −5 ◦C
(4.75)
(a),(b) z∗ ζ
(a) (c)
a1 −5 ◦C Ferrier 12.96b1 −5 ◦C Ferrier 0.639B Huffmann and Vail 4.5mi0 10−12 kgNi0 Fletcher 10−2 m−3
Ni01 −5 ◦C Ferrier 103 m−3
Ni02 −5 ◦C Ferrier 50 m−3
qvsi kg kg−1
qvsw kg kg−1
T KT0 273.16 KTs T0 − T KSi
Swi
SSi Si − 1w z m s−1
β2 Fletcher 0.6 K−1
ρ kg m−3
8Ferrier (1994)
58 4
2 NUFci
Bigg (1953) 9
NUFci = B′ [exp (A′Ts) − 1]ρq2
c
ρwNc(4.76)
NUFNci = B′ [exp (A′Ts) − 1]
qc
ρw(4.77)
A′ Bigg 0.66 K−1
B′ Bigg 100.0 m−3 s−1
Nc 1×108 m−3
T KT0 273.16 KTs T0 − T Kρ kg m−3
ρw 1×103 kg m−3
3 NUCci
10 3
• Brownian diffusion
• diffusiophoresis
• thermophoresis
[dNc
dt
]b
= F1Dar (4.78)
[dNc
dt
]v
= F1F2RvT
Lv(4.79)
[dNc
dt
]t
= F1F2ft (4.80)
9 (1999), Ikawa and Saito (1991)10 (1999), Cotton et al. (1986), Young (1974)
4.2 59
F1 = 2πDcNcNar (4.81)
F2 =κa
p(T − Tcl) (4.82)
ft =0.4 [1 + 1.45Kn + 0.4 exp (−1 /Kn )] (κ + 2.5Knκa)
(1 + 3Kn) (2κ + 5κaKn + κa)(4.83)
Kn Knudsen
Kn = λa0p00
T00Ra
T
p(4.84)
Dar
Dar =kTcl
6πRaµ(1 + Kn) (4.85)
Tcl Nar11
Nar = Na0 (270.16 − Tcl)1.3 (4.86)
NUCNci =
1ρ
([dNc
dt
]b
+[dNc
dt
]v
+[dNc
dt
]t
)(4.87)
NUCci =ρqc
NcNUCN
ci (4.88)
3
Dc mk 1.380658×10−23 J K−1
Lv J kg−1
Na0 2×105 m−3
Nc 1×108 m−3
11Cotton et al. (1986)
60 4
p Pap00 101325 PaRa 3×10−7 mRv 461.0 J K−1 kg−1
T KT00 293.15 KTcl Kκ 2.4 × 10−2 J m−1 s−1 K−1
κa J m−1 s−1 K−1
λa0 p00, T00 6.6×10−8 mµ kg m−1 s−1
ρ kg m−3
4 NUHci
−40 ◦C 12
NUHNci =
1ρ
Nc
2∆t(4.89)
NUHci =qc
2∆t(4.90)
2∆t
Ferrier,1994
2 SP
2
• Hallett and Mossop, 1974
• Vardiman, 1978
• Hobbs and Rangno, 1985
1 Hallett-Mossop rime splintering mechanism13 2
SPNsi =
1ρ× 3.5 × 108f (Ts) CLcs (4.91)
SPsi = mi0SPNsi (4.92)
12Ikawa and Saito (1991), Ferrier (1994)13 (1999), Ikawa and Saito (1991), Cotton et al. (1986)
4.2 61
SPNgi =
1ρ× 3.5 × 108f (Tg)CLcg (4.93)
SPgi = mi0SPNgi (4.94)
2f (Tx) Tx Ts Tg
f (Tx) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
0, Tx > 270.16 K
Tx − 268.162
, 268.16 ≤ Tx ≤ 270.16 K
268.16 − Tx
3, 265.16 ≤ Tx ≤ 268.16 K
0, Tx < 265.16 K
(4.95)
Cotton et al. (1986) (72) f (Tx) (4.95)(4.95) Tx = 268.16 K f (Tx) = 0 Ikawa et al. (1991), Ikawa
and Saito (1991)
f (Tx) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
0, Tx ≥ 270.16 K
270.16 − Tx
2, 268.16 < Tx < 270.16 K
1, Tx = 268.16 K
Tx − 265.163
, 265.16 ≤ Tx < 268.16 K
0, Tx < 265.16 K
(4.96)
Tx = 268.16 K f (Tx) = 1 −5 ◦C
CLcg s−1
CLcs s−1
mi0 10−12 kgTs KTg KTx Kρ kg m−3
62 4
V D
→→
→→ V Dvr < 0
→ V Dvi > 0→ V Dvs > 0→ V Dvg > 0→ V Dvh > 0
→ V Dvi < 0→ V Dvs < 0→ V Dvg < 0→ V Dvh < 0
1 V Drv
V Dvr = −V Drv =
⎧⎪⎪⎨⎪⎪⎩
2π
ρ(Sw − 1)Gw (T, p)V ENTr, Sw − 1 < 0
0, Sw − 1 ≥ 0
(4.97)
14
Gw (T, p) =( L2
v
κRvT 2+
1ρqvswDv
)−1
(4.98)
V ENTr = nr0
[0.78λ−2
r + 0.31S13c ν− 1
2 α12urΓ
(5 + βur
2
)λ− (5+βur)
2r
(ρ0
ρ
) 14]
(4.99)
14 (1999)
4.2 63
Dv m2 s−1
Lv J kg−1
nr0 y 8.0 × 106 m−4
qvsw kg kg−1
Rv 461.0 J K−1 kg−1
Sc 0.6Sw − 1T Kαur 842 m1−βur s−1
βur 0.8λr m−1
κ 2.4×10−2 J m−1 s−1 K−1
ν m2 s−1
ρ kg m−3
ρ0 kg m−3
2 V Dvs, V Dvg15
x = s, g
T < T0
V Dvx =2π
ρ(Si − 1)Gi (T, p) V ENTx − LsLf
κRvT 2Gi (T, p)CLcx (4.100)
T > T0 MLxr < 0 MLxr ≥ 0
V Dvx =
⎧⎪⎪⎨⎪⎪⎩
2π
ρ(Sw − 1) Gw (T, p)V ENTx, MLxr ≥ 0
2πDv (qv − qvs (T0))V ENTx, MLxr < 0
(4.101)
Gi (T, p) =( L2
s
κRvT 2+
1ρqvsiDv
)−1
(4.102)
15 (1999), Ikawa and Saito (1991), Lin et al (1983)
64 4
Gw (T, p) (4.98) x = s, g
V ENTx = nx0
[0.78λ−2
x + 0.31S13c ν− 1
2 α12uxΓ
(5 + βux
2
)λ− (5+βux)
2x
(ρ0
ρ
) 14]
(4.103)
CLcg s−1
CLcs s−1
Dv m2 s−1
Lf J kg−1
Ls J kg−1
ng0 y m−4
ns0 y m−4
MLgr s−1
MLsr s−1
qvs (T0) kg kg−1
qvsi kg kg−1
Rv 461.0 J K−1 kg−1
Sc 0.6Si − 1Sw − 1T KT0 273.16 Kαug 124 m1−βug s−1
αus 17 m1−βus s−1
βug 0.64βus 0.5λg m−1
λs m−1
κ 2.4×10−2 J m−1 s−1 K−1
ν m2 s−1
ρ kg m−3
ρ0 kg m−3
3 V Dvi16
V Dvi =qv − qvsi
qvsw − qvsia1 (mi)
a2 Ni
ρ(4.104)
16Ikawa and Saito (1991), Ikawa et al. (1991)
4.2 65
mi
mi =qiρ
Ni(4.105)
a1, a2 Koenig (1971) 1◦C
a1
Tc◦C 0 -10 -20 -30
0.0 0.000 7.434 ×10−10 9.115 ×10−10 5.333 ×10−10
< -1.0 7.939 ×10−11 1.812 ×10−09 4.876 ×10−10 4.834 ×10−10
< -2.0 7.841 ×10−10 4.394 ×10−09 3.473 ×10−10
< -3.0 3.369 ×10−09 9.145 ×10−09 4.758 ×10−10
< -4.0 4.336 ×10−09 1.725 ×10−10 6.306 ×10−10
< -5.0 5.285 ×10−09 3.348 ×10−08 8.573 ×10−10
< -6.0 3.728 ×10−09 1.725 ×10−08 7.868 ×10−10
< -7.0 1.852 ×10−09 9.175 ×10−09 7.192 ×10−10
< -8.0 2.991 ×10−10 4.412 ×10−09 6.153 ×10−10
< -9.0 4.248 ×10−10 2.252 ×10−09 5.956 ×10−10
a2
Tc◦C 0 -10 -20 -30
0.0 0.000 4.318 ×10−01 4.447 ×10−01 4.382 ×10−01
< -1.0 4.006 ×10−01 4.771 ×10−01 4.126 ×10−01 4.361 ×10−01
< -2.0 4.831 ×10−01 5.183 ×10−01 3.960 ×10−01
< -3.0 5.320 ×10−01 5.463 ×10−01 4.149 ×10−01
< -4.0 5.307 ×10−01 5.651 ×10−01 4.320 ×10−01
< -5.0 5.319 ×10−01 5.813 ×10−01 4.506 ×10−01
< -6.0 5.249 ×10−01 5.655 ×10−01 4.483 ×10−01
< -7.0 4.888 ×10−01 5.478 ×10−01 4.460 ×10−01
< -8.0 3.894 ×10−01 5.203 ×10−01 4.433 ×10−01
< -9.0 4.047 ×10−01 4.906 ×10−01 4.413 ×10−01
qvsi kg kg−1
qvsw kg kg−1
Tc◦C
ρ kg m−3
66 4
CL
qc qi CLci
qc qs CLcs
qc qg CLcg
qc qr CLcr
qr qi CLri
qr qs CLrs
qr qg CLrg
qi qr CLir
qi qs CLis
qi qg CLig
qs qr CLsr
qs qg CLsg
17
1 CLxy x, y = r, s, g; x �= y
CLxy = π2 ρx
ρExy
√(Ux − Uy
)2 + αUxUynx0ny0
(5
λ6xλy
+2
λ5xλ2
y
+0.5
λ4xλ3
y
)(4.106)
CLNxy =
π
2ρExy
√(Ux − Uy
)2 + αUxUynx0ny0
(1
λ3xλy
+1
λ2xλ2
y
+1
λxλ3y
)(4.107)
x, y = r, s, g; x �= y
Exy
nx0 x y m−4
Ux x m s−1
α 0.04λx x m−1
ρ kg m−3
ρx x kg m−3
17 (1999), Lin et al. (1983), Murakami (1990), Ikawa and Saito (1991)
4.2 67
2 CLcy, CLiy y = r, s, g
CLxy =π
4Exy ny0 qx αuy Γ (3 + βuy) λ−(3+βuy)
y
(ρ0
ρ
) 12
(4.108)
Eiy
Ecy
Ecy =Stk2
(Stk + 0.5)2(4.109)
Stk Ikawaand Saito (1991)
Stk = D2cρw
Uy
9µDy(4.110)
Dc mDy y mny0 y y m−4
Uy y m s−1
αuy y m1−βuy s−1
βuy y
λy y m−1
µ kg m−1 s−1
ρ kg m−3
ρ0 kg m−3
ρw 1×103 kg m−3
3 CLri
18
CLri =π2
24Eir Ni nr0 αur Γ (6 + βur) λ−(6+βur)
r
(ρ0
ρ
) 12
(4.111)
18 (1999)
68 4
CLNri =
π
4ρEir Ni nr0 αur Γ (3 + βur) λ−(3+βur)
r
(ρ0
ρ
) 12
(4.112)
Eir 1.0nr0 y 8.0 × 106 m−4
αur 842 m1−βur s−1
βur 0.8λr m−1
ρ kg m−3
ρ0 kg m−3
4 Exy19
(1) (3) Exy
Ecr Stk2/
(Stk + 0.5)2
Ecs Stk2/
(Stk + 0.5)2
Ecg Stk2/
(Stk + 0.5)2
Ers 1.0Erg 1.0Eir 1.0Eis 1.0Eig 0.1Esr 1.0Esg 0.001
5 αrs
0 ◦C(4.31) (4.32) αrs mr ms
αrs =m2
s
m2s + m2
r
(4.113)
mr ms
mr = ρr
(4λr
)3
(4.114)
19Ikawa and Saito (1991), Ikawa et al. (1991)
4.2 69
ms = ρs
(4λs
)3
(4.115)
(1 − αrs)(4.113)
PG
PGdry = CLcg + CLrg + CLig + CLsg (4.116)
20
PGwet =2π [κTs + LvDvρ (qvs (T0) − qv)]
ρ (Lf − CwTs)V ENTg +
(CL′
ig + CL′sg
) (1 +
CiTs
Lf − CwTs
)(4.117)
V ENTg (4.103)PGdry PGwet
PGg = PGdry, PGdry ≤ PGwet (4.118)
PGg = PGwet, PGdry > PGwet (4.119)
CLcg s−1
CLig s−1
CL′ig s−1
CLrg s−1
CLsg s−1
20 (1999)
70 4
CL′sg s−1
Ci 2.0×103 J K−1kg−1
Cw 4.17×103 J K−1kg−1
Dv m2 s−1
Lf J kg−1
Lv J kg−1
qvs (T0) kg kg−1
T KT0 273.16 KTs T0 − T Kκ 2.4×10−2 J m−1 s−1 K−1
ρ kg m−3
AG
2
1 AGNi
21
AGNi =
[d
dt
(Ni
ρ
)]aggr
= − c1
2ρNi (4.120)
c1
c1 =ρqiαuiEiiX
ρi
(ρ0
ρ
) 13
(4.121)
Eii 0.1X 0.25αui 700 m1−βui s−1
βui 1.0ρ kg m−3
ρ0 kg m−3
ρi 5.0×102 kg m−3
21 (1999), Ikawa and Saito (1991)
4.2 71
2 AGNs
22
qs
Ns
AGNs =
[d
dt
(Ns
ρ
)]aggr
= −1ρ
αusEssI (βus)4 × 720
π1−βus
3 ρ2+βus
3 ρ−2−βus
3s q
2+βus3
s N4−βus
3s (4.122)
I (βus) =∫ ∞
0
∫ ∞
0
x3y3 (x + y)2∣∣xβ
us − yβus
∣∣ exp [− (x + y)] dxdy (4.123)
Gauss 23
I (βus) = Γ (βus) 21−d3∑
i=1
Ci
[F (1, d; 8 − i; 0.5)
7 − i− F (1, d; 4 + βus; 0.5)
3 + βus + i
](4.128)
d = 10 + βus
C1 = 1
C2 = 3
C3 = 1
I (βus) Ikawa and Saito, 1991; Mizuno, 1990
22 (1999), Ikawa and Saito (1991)23 hypergeometric function 2 x = 0, 1,∞1
F (x, a; b; c) = 1 +a · b
c
x
1!+
a (a + 1) b (b + 1)
c (c + 1)
x2
2!+ · · · (4.124)
c �= 0,−1,−2, · · ·
(a)n = a (a + 1) (a + 2) · · · (a + n − 1) =(a + n − 1)!
(a − 1)!(4.125)
(a)0 = 1 (4.126)
F (x, a; b; c) =
∞∑n=0
(a)n (b)n
(c)n
xn
n!(4.127)
72 4
βus 0.4 0.5 0.6
I (βus) 1108 1610 2566
Ess 0.1αus 17 m1−βus s−1
βus 0.5ρ kg m−3
ρs 8.4×101 kg m−3
CN
CNxy Kessler (1969)
qc qr CNcr
qi qs CNis
qs qg CNsg
qs qh CNsh
qg qh CNgh
qs qg CNsg
qs qh CNsh
qg qh CNgh
CNcr, CNis, CNsg
1 CNcr24
Kessler (1969) Berry (1968), Berry and Reinhardt (1974)
a Berry 1968 , Berry and Reinhardt 1974 25
Berry (1968), Berry and Reinhardt (1974)
CNcr =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
0.104gEcc
µ (Ncρw)13
(ρ4q7
c
) 13 , qc ≥ qcm
0, qc < qcm
(4.129)
24 (1999), Lin et al. (1983), Ferrier (1994), Ikawa and Saito (1991)25 (1991)
4.2 73
Ecc = 0.55qcm
qcm =ρw
6ρπD3
cmNc (4.130)
Dcm Dcm = 20 µm Nc
Nc = 108 m−3
b Kessler 1969 26
CNcr = a (qc − qcm)H (qc − qcm) (4.131)
H a = 10−3 s−1 , qcm = 10−3 kg kg−1 Cottonand Anthes (1989) a qcm qc
a = πEccUdcNcD2c = 1.3 × q
43c N
− 13
c
(ρ0
ρ
)(4.132)
qcm =4πρwNcD
3cm
3ρ= 4 × 10−12Nc, Dcm = 10−5 m (4.133)
c Lin et al. 1983
Berry (1968) Lin et al. (1983)
CNcr = ρ (qc − qcm)2[1.2 × 10−4 + 1.569 × 10−12 Nc
σ2(qc−qcm)
](4.134)
σ2 = 0.15 qcm = 2 × 10−3 kg kg−1
(a) (c)
g 9.8 m s−2
Nc 1×108 m−3
µ kg m−1 s−1
ρ kg m−3
ρ0 kg m−3
ρw 1×103 kg m−3
26Ikawa and Saito (1991)
74 4
2 CNis
227
Ri Rs0 ∆tis1
∆tis1 =R2
s0 − R2i
2a1ρi (4.135)
a1 (4.100) (4.97)
a1 = (Si − 1)( L2
s
κRvT 2+
1ρqvsiDv
)−1
(4.136)
CNdepis
CNdepis =
qi
∆tis1(4.137)
Ri Rs0 ∆tis2
ρi = const. Ni Ni (Ri /Rs0 )3
∆tis2 =2c1
log(
Rs0
Ri
)3
(4.138)
c1 (4.121)
CNaggis =
qi
∆tis2(4.139)
CNis
CNis = CNdepis + CNagg
is (4.140)
27 (1999), Murakami (1990), Ikawa and Saito (1991)
4.2 75
Dv m2 s−1
Ls J kg−1
qvsi kg kg−1
Rv 461.0 J K−1 kg−1
Si − 1T Kκ 2.4×10−2 J m−1 s−1 K−1
ρ kg m−3
ρi 5.0×102 kg m−3
3 CNsg
riming embryo 28 riming
CN rimsg =
3πρ0 (ρqc)2 E2
csα2usΓ (2βus + 2)
8ρ (ρg − ρs)λ2βus+1s
Ns (4.141)
embryo
CNembsg =
ρs
ρg − ρs
3πρ0 (ρqc)2 E2
csα2usΓ (2βus + 2)
8ρ (ρg − ρs)λ2βus+1s
Ns (4.142)
CNsg
CNsg = CN rimsg + CNemb
sg
=ρg
ρg − ρs
3πρ0 (ρqc)2E2
csα2usΓ (2βus + 2)
8ρ (ρg − ρs)λ2βus+1s
Ns (4.143)
CNNsg =
ρ0
ρ
[3παusEcsρqc
2 (ρg − ρs)
]Ns (4.144)
28Murakami (1990)
76 4
Ecs Stk2/
(Stk + 0.5)2
αus 17 m1−βus s−1
βus 0.5λs m−1
ρ kg m−3
ρ0 kg m−3
ρg 3.0×102 kg m−3
ρs 8.4×101 kg m−3
ML
1 MLic
T > T0
Tc > T0
MLic =qi
2∆t(4.145)
2∆t
2 MLsr, MLgr
MLxr x = s, g 29
MLxr =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
2π
ρLf[κTc + LvDvρ (qv − qvs (T0))]V ENTx +
CwTc
Lf(CLcx + CLrx) , T > T0
0, T ≤ T0
(4.146)
T > T0 MLxr < 0 MLxr = 0V ENTx (4.103)
(1),(2)
CLcx x s−1
CLrx x s−1
Cw 4.17×103 J K−1 kg−1
29Ikawa and Saito (1991)
4.2 77
Dv m2 s−1
Lf J kg−1
Lv J kg−1
qvs (T0) kg kg−1
T KT0 273.16 KTc
◦Cκ 2.4×10−2 J m−1 s−1 K−1
ρ kg m−3
FR
FRrg Bigg (1953) 30
FRrg = 20π2B′nr0ρw
ρ[exp (A′Ts) − 1]λ−7
r (4.147)
FRNrg =
π
6ρB′nr0 [exp (A′Ts) − 1]λ−4
r (4.148)
A′ Bigg 0.66 K−1
B′ Bigg 100.0 m−3 s−1
nr0 y 8.0 × 106 m−4
T KT0 273.16 KTs T0 − T Kλr m−1
ρ kg m−3
ρw 1×103 kg m−3
30Lin et al. (1983), (1999)
78 4
SH
T > T0
SHsr = CLcs + CLrs (4.149)
SHgr = CLcg + CLrg (4.150)
T ≤ T0
SHgr = CLcg + CLrg + CL′ig + CL′
sg − PGwet (4.151)
Ferrier (1994)
CLcg s−1
CLcs s−1
CLrg s−1
CLrs s−1
CL′ig s−1
CL′sg s−1
PGwet s−1
T KT0 273.16 K
8mm
λr
4.2 79
4.2.5
31
∗
∆qc = q∗v − q∗vsw (4.152)
∆qc > 0 q∗c > 0
θt+∆t = θ∗ + γ (q∗v − q∗vsw)/(
1 + γ∂q∗vsw
∂θ∗
)(4.153)
qt+∆tv = q∗v +
(θ∗ − θt+∆t
)/γ (4.154)
qt+∆tc = q∗v + q∗c − qt+∆t
v (4.155)
θ, qv, qc qt+∆tc > 0 ∗
(4.153) (4.155)γ ≡ Lv/ (CpΠ)
qt+∆tc ≤ 0
θt+∆t = θ∗ − γq∗c (4.156)
qt+∆tv = q∗v + q∗c (4.157)
qt+∆tc = 0 (4.158)
Cp 1004 J K kg−1
Lv J kg−1
qvsw kg kg−1
Π
31Soong and Ogura (1973)
80 4
4.2.6
Fall.qx =1ρ
∂ρUxqqx
∂z(4.159)
x Uxq (4.68)
Fall.Nx
ρ=
1ρ
∂NxUxN
∂z(4.160)
x UxN (4.67)
CFL∆tlim
∆tlim =∆z
Uxq(4.161)
2∆t
∆tfall =2∆t
int (2∆t/∆tlim) + 1, int (4.162)
CFL 1
z∗ ζ
4.2.7
qvsw, qvsi32
qvsw = ε610.78
pexp
(17.269
T − T0
T − 35.86
)kg kg−1 (4.163)
qvsi = ε610.78
pexp
(21.875
T − T0
T − 7.86
)kg kg−1 (4.164)
32Orville and Kopp (1977), Murray (1966)
4.2 81
Lv,Ls,Lf
Lv = 2.50078 × 106
(T0
T
)(0.167+3.67×10−4T)J kg−1 (4.165)
Ls = 2.834 × 106 + 100 (T − T0) J kg−1 (4.166)
Lf = 3.34 × 105 + 2500 (T − T0) J kg−1 (4.167)
ν, µ
ν = 1.328 × 10−5 p0
p
(T
T0
)1.754
m2 s−1 (4.168)
µ = ρν kg m−1 s−1 (4.169)
Dv
Dv = 2.23 × 10−5 p0
p
(T
T0
)1.81
m2 s−1 (4.170)
p Pap0 101325 PaT KT0 273.16 Kε 0.622ρ kg m−3